Solution of the Schrödinger equation with the use of the translation operator

И. Д. Феранчук, L I Komarov

Boris V. Alekseev, A. I. Аbakumov

Francesco R. Ruggeri

A process exists in the literature (1) of creating an information term based on Fisher’s information using spatial density P and adding it to a classical Lagrangian based on a velocity field i.e. velocity=dS/dx. Maximization of S and P is performed and a differential equation in terms of spatial density and S is obtained. Finally, it is noted that a change of variables: W= sqrt(p) exp(iS/hbar) may simplify the differential equation so that it becomes the Schrodinger equation with W as the wavefunction. In this note, we consider the wavefunction as appearing “naturally” as a normalization constant when using conditional probability and argue that W is the velocity field. Given that the Schrodinger equation includes a term of the form -1/2m d/dx d/dx W (in 1-D) i.e. the first derivative is related to the average velocity and the second to the ensemble average of p*p. This follows because unnormalized conditional probability f(p,x) is the Fourier basis function i.e. exp(ipx). The second derivative term - 1/2m d/dx d/dx W may be obtained from a Lagrangian containing (d/dxW) (d/dx W), thus formally one may use maximization to obtain the Schrodinger equation at a level using W directly. The maximization form, however, is identical to the “classical translational” form (2) of Fisher’s information. Thus, it seems that the Fourier basis exp(ipx) and the maximization of classical translational Fisher’s information are linked. In this note, we wish to investigate this in further detail. We also use translational properties together with conditional probability to show that f(p,x) where P(p/x)= a(p) f(p/x) / [Sum over p a(p) f(p/x)] is exp(ipx). In particular, we obtain an equation taken from the translation of the conservation of energy equation at point x to x+delta x. This links df(p,x)/dx with -dV/dx (the force). If the force is stochastic and yields “k” momentum hits, -dV/dx= Sum over k kVk(x). We argue that df(p,x)/dx also brings out this k term i.e. is an eigenfunction of k i.e. -idf(p,x)/dx =k f(p,x) yields exp(ipx) as a solution. This f(p,x) is automatically an orthonormal basis (multiplied by an appropriate constant) and helps to show why the “momentum operator” is linked to a “translation operator” i.e. d/dx. The bound state Schrodinger equation is considered in this note.

Boris V. Alekseev, A. I. Аbakumov

Richard L. Hall

Continuation of previous work (ibid., vol.11, no.7, p.1227 (1978)). Earlier exact solutions of Schrodinger's equation for translation-invariant systems of particles interacting by Hooke's law pair potentials are augmented to include systems consisting of an arbitrary number S of groups of identical particles. Exact solutions can always be found whenever the number of distinct masses plus the number of distinct coupling constants does not exceed (2S+1). The case S=2 is solved in detail and is applied to show that harmonic matter is never stable.